Going Down the Exponential Slide

A bead slides under the pull of gravity, g,g, down a frictionless wire segment in the shape of the curve y=exy = e^{-x}, where xx is the horizontal direction and yy is the vertical direction. The bead starts from rest at (x,y)=(0,1)(x,y) = (0,1).

The time it takes for the particle to travel between x=ax=a and x=bx=b can be expressed as

ta,b=12gab1+PeQx1+ReSxdx,\large{t_{a,b} = \frac{1}{\sqrt{2g}} \int_a^b \sqrt{\frac{1 + P e^{Q x}}{1 + R e^{S x} }}\,dx},

where P,Q,R,P,Q,R, and SS are integers.

Determine P+Q+R+S.P + Q + R + S.

Note: The constant ee is Euler's number.

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